AMSTERDAM, May 22 (Reuters) - Akzo Nobel shareholders angered by the Dulux paint maker's rejection of a 26.3 billion euro ($29.5 billion) takeover proposal from U.S. rival PPG Industries took their fight to an Amsterdam court on Monday.

Activist hedge fund Elliott Advisors, supported by several long-term institutional investors, asked the Amsterdam Enterprise Chamber to order an investigation into possible mismanagement by Akzo's board and force an extraordinary meeting of shareholders to vote on dismissing Chairman Antony Burgmans.

Elliott Advisors and the institutional investors together represent 18 percent of the Dutch company's shares.

"A large group of shareholders has lost confidence in Mr. Burgmans and has asked to call him to account at an extraordinary shareholders meeting," said Jan Willem de Groot, representing Elliott.

"That's a vote of no confidence by itself," he told the court hearing attended by both Burgmans and PPG Chief Executive Michael McGarry.

Akzo lawyer Jan de Bie Leuveling Tjeenk responded that under Dutch law, company boards, not shareholders, have the right to determine strategy.

He said Akzo's boards acted unanimously in rejecting PPG's advances and repeated twice for emphasis that Burgmans' dismissal would "change nothing."

As more than 100 lawyers and journalists packed the courtroom and public gallery for the hearing in the high-stakes corporate battle, PPG boss McGarry shook hands with Burgmans.

Both later briefly addressed the court, with McGarry saying PPG was still prepared to enter talks with Burgmans, and Burgmans saying PPG was free to make offers but Akzo Nobel was not obliged to enter negotiations.

Judges said they expected to rule after the stock market closes on May 29 - just in time for PPG to decide whether it wants to submit a formal bid to Dutch regulators without the support of Akzo's board by a June 1 deadline, or walk away for at least six months.

Despite impassioned pleas by several shareholders, experts in Dutch corporate law say it will be tough to convince judges that Akzo's governance has been so poor as to warrant an investigation.

Shares in Akzo closed up 0.5 percent at 75.70 euros on Monday, well below PPG's 96.75 euros per share bid proposal made on April 20, suggesting investors are sceptical a PPG offer will ultimately succeed.

Akzo has argued a PPG takeover would be bad for employees, that the companies' cultures don't mesh, that a deal faces antitrust risks, that it would be bad for the environment and that Akzo should stay Dutch in the country's national interest.

Several shareholders are openly sceptical of those arguments and question the company's judgment in declining talks on PPG's proposal.

Presiding Judge Gijs Bekkink questioned PPG lawyers over whether the company shouldn't have better addressed Akzo's objections, and lawyers for Akzo on why it had spurned talks.

PPG lawyer Arnold Croiset van Uchelen said the company had tried its best to address Akzo's concerns in each of the three proposals it has made since its pursuit of Akzo began in March.

"To be more concrete, you need talks and due diligence," he said. "Akzo is always vague about where exactly the problem is."

Akzo lawyer Fons Leijten said entering talks would quickly have raised expectations of a deal. "They always come with strings attached," he said.